On the extremal cacti with minimum Sombor index

Let H be a graph with edge set E-H. The Sombor index and the reduced S O(H) = Sigma(uvEEH)root d(H)(u)(2) + d(H)(v)(2) and S Ored(H) = root(d(H)(u) - 1)(2) + (d(H)(v) - 1)(2), respectively Where d(H)(u) and d(H)(v) are the degrees of the vertices u and v in H, respectively. A cactus is a connected graph in which any two cycles have at most one common vertex. Let C(n, k) be the class of cacti of order n with k cycles. In this paper, the lower bound for the Sombor index of the cacti in C(n, k) is obtained and the corresponding extremal cacti are characterized when n >= 4k - 2 and k >= 2. Moreover, the lower bound of the reduced Sombor index of cacti is obtained by similar approach.